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This book is an introduction to the contemporary representation theory of Artin algebras, by three very distinguished practitioners in the field. Beyond assuming some first-year graduate algebra and basic homological algebra, the presentation is entirely self-contained, so the book is a suitable introduction for any mathematician (especially graduate students) to this field. The main aim of the book is to illustrate how the theory of almost split sequences is used in the representation theory of Artin algebras. However, other foundational aspects of the subject are developed. These results give concrete illustrations of some of the more abstract concepts and theorems. The book includes complete proofs of all theorems, and numerous exercises.

Artin rings. --- Artin algebras. --- Representations of algebras.

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Ordered algebraic structures --- Artin rings. --- Artin algebras. --- Representations of algebras.

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This is a monograph which collects basic techniques, major results and interesting applications of Lefschetz properties of Artinian algebras. The origin of the Lefschetz properties of Artinian algebras is the Hard Lefschetz Theorem, which is a major result in algebraic geometry. However, for the last two decades, numerous applications of the Lefschetz properties to other areas of mathematics have been found, as a result of which the theory of the Lefschetz properties is now of great interest in its own right. It also has ties to other areas, including combinatorics, algebraic geometry, algebraic topology, commutative algebra and representation theory. The connections between the Lefschetz property and other areas of mathematics are not only diverse, but sometimes quite surprising, e.g. its ties to the Schur-Weyl duality. This is the first book solely devoted to the Lefschetz properties and is the first attempt to treat those properties systematically.

Geometry, Algebraic --- Artin rings --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Mathematical Theory --- Artinian rings --- Rings, Artin --- Rings, Artinian --- Algebraic geometry --- Mathematics. --- Algebra. --- Algebraic geometry. --- Combinatorics. --- Algebraic Geometry. --- Geometry, Algebraic. --- Artin rings. --- Associative rings --- Commutative rings --- Geometry --- Geometry, algebraic. --- Combinatorics --- Mathematical analysis

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Highlights developments on artinian modules over group rings of generalized nilpotent groups. This work includes traditional topics such as direct decompositions of artinian modules, criteria of complementability, and criteria of semisimplicity of artinian modules. It also allows a generalization of Maschke Theorem in classes of infinite groups.

Artin rings --- Group rings --- Nilpotent groups. --- Anneaux artiniens --- Anneaux de groupes --- Groupes nilpotents --- Artin rings. --- Group rings. --- Nilpotent groups --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Groups, Nilpotent --- Artinian rings --- Rings, Artin --- Rings, Artinian --- Mathematics. --- Algebra. --- Finite groups --- Group theory --- Rings (Algebra) --- Associative rings --- Commutative rings --- Mathematical analysis

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Ordered algebraic structures --- Artin algebras. --- Algèbres artiniennes. --- Gorenstein rings. --- Anneaux de Gorenstein. --- Characteristic functions. --- Fonctions caractéristiques. --- Artin algebras --- Characteristic functions --- Gorenstein rings --- Gorenstein's rings --- Rings, Gorenstein --- Noetherian rings --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Probabilities --- Algebras, Artin --- Artin rings --- Commutative algebra --- Modules (Algebra)